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Abstract

In this article, we investigate the multiplicity of positive solutions for a fourth-order
system of integral boundary value problem on time scales. The existence of multiple
positive solutions for the system is obtained by using the fixed point theorem of
cone expansion and compression type due to Krasnosel'skill. To demonstrate the applications
of our results, an example is also given in the article.

Keywords:

1 Introduction

Boundary value problem (BVP) for ordinary differential equations arise in different
areas of applied mathematics and physics and so on, the existence and multiplicity
of positive solutions for such problems have become an important area of investigation
in recent years, lots of significant results have been established by using upper
and lower solution arguments, fixed point indexes, fixed point theorems and so on
(see [1-8] and the references therein). Especially, the existence of positive solutions of nonlinear
BVP with integral boundary conditions has been extensively studied by many authors
(see [9-18] and the references therein).

However, the corresponding results for BVP with integral boundary conditions on time
scales are still very few [19-21]. In this article, we discuss the multiple positive solutions for the following fourth-order
system of integral BVP with a parameter on time scales

The main purpose of this article is to establish some sufficient conditions for the
existence of at least two positive solutions for system (1.1) by using the fixed point
theorem of cone expansion and compression type. This article is organized as follows.
In Section 2, some useful lemmas are established. In Section 3, by using the fixed
point theorem of cone expansion and compression type, we establish sufficient conditions
for the existence of at least two positive solutions for system (1.1). An illustrative
example is given in Section 4.

2 Preliminaries

In this section, we will provide several foundational definitions and results from
the calculus on time scales and give some lemmas which are used in the proof of our
main results.

A time scale is a nonempty closed subset of the real numbers ℝ.

Definition 2.1. [22]For , we define the forward jump operator by , while the backward jump operator by .

In this definition, we put and , where ∅, denotes the empty set. If σ(t) > t, we say that t is right-scattered, while if ρ(t) < t, we say that t is left-scattered. Also, if and σ(t) = t, then t is called right-dense, and if and ρ(t) = t, then t is called left-dense. We also need, below, the set , which is derived from the time scale as follows: if has a left-scattered maximum m, then . Otherwise, .

Definition 2.2. [22]Assume that is a function and let . Then x is called differentiable at if there exists a θ ∈ ℝ such that for any given ε > 0, there is an open neighborhood U of t such that

In this case, xΔ(t) is called the delta derivative of x at t. The second derivative of x(t) is defined by xΔΔ(t) = (xΔ)Δ(t).

In a similar way, we can obtain the fourth-order derivative of x(t) is defined by x(4Δ)(t) = (((xΔ)Δ)Δ)Δ(t).

Definition 2.3. [22]A function is called rd-continuous provided it is continuous at right-dense points in and its left-sided limits exist at left-dense points in . The set of rd-continuous functions will be denoted by .

Definition 2.4. [22]A function is called a delta-antiderivative of provide FΔ(t) = f(t) holds for all . In this case we define the integral of f by

For convenience, we denote , and for i = 1, 2, we set

where

To establish the existence of multiple positive solutions of system (1.1), let us
list the following assumptions:

In order to overcome the difficulty due to the dependence of f, g on derivatives, we first consider the following second-order nonlinear system

(2.1)

where A0 is the identity operator, and

(2.2)

For the proof of our main results, we will make use of the following lemmas.

Lemma 2.1. The fourth-order system (1.1) has a solution (x, y) if and only if the nonlinear system (2.1) has a solution (u, v).

Proof. If (x, y) is a solution of the fourth-order system (1.1), let u(t) = xΔΔ(t), v(t) = yΔΔ(t), then it follows from the boundary conditions of system (1.1) that

Thus (u, v) = (xΔΔ(t), yΔΔ(t)) is a solution of the nonlinear system (2.1).

Conversely, if (u, v) is a solution of the nonlinear system (2.1), let x(t) = A2u(t), y(t) = A2v(t), then we have

which imply that

Consequently, (x, y) = (A2u(t), A2v(t)) is a solution of the fourth-order system (1.1). This completes the proof.

Lemma 2.2. Assume that D11D21 ≠ 1 holds. Then for any h1 ∈ C(I', ℝ+), the following BVP

(2.3)

has a solution

where

Proof. First suppose that u is a solution of system (2.3). It is easy to see by integration of BVP(2.3) that

(2.4)

Integrating again, we can obtain

(2.5)

Let t = σ(T) in (2.4) and (2.5), we obtain

(2.6)

(2.7)

Substituting (2.6) and (2.7) into the second boundary value condition of system (2.3),
we obtain

(2.8)

From (2.8) and the first boundary value condition of system (2.3), we have

(2.9)

(2.10)

Substituting (2.9) and (2.10) into (2.5), we have

(2.11)

By (2.11), we get

(2.12)

(2.13)

By (2.12) and (2.13), we get

(2.14)

(2.15)

Substituting (2.14) and (2.15) into (2.11), we have

(2.16)

Conversely, suppose , then

(2.17)

Direct differentiation of (2.17) implies

and

and it is easy to verify that

This completes the proof.

Lemma 2.3. Assume that D12D22 ≠ 1 holds. Then for any h2 ∈ C(I', ℝ+), the following BVP

has a solution

where

Proof. The proof is similar to that of Lemma 2.2 and will omit it here.

Lemma 2.4. Suppose that (H1) is satisfied, for all t, s ∈ I and i = 1, 2, we have

On the one hand, from the definition of Li and mi, for all t, s ∈ I, we have

and on the other hand, we obtain easily that from the definition of Mi, for all t, s ∈ I,

Finally, it is easy to verify that mGi(σ(s), s) ≤ Hi(t, s) ≤ MGi(σ(s), s). This completes the proof.

Lemma 2.5. [23]Let E be a Banach space and P be a cone in E. Assume that Ω1 and Ω2 are bounded open subsets of E, such that 0 ∈ Ω1, , and let be a completely continuous operator such that either

(i) ||Tu|| ≤ ||u||, ∀u ∈ P ∩ ∂Ω1 and ||Tu|| ≥ ||u||, ∀u ∈ P ∩ ∂Ω2, or

(ii) ||Tu|| ≥ ||u||, ∀u ∈ P ∩ ∂Ω1 and ||Tu|| ≤ ||u||, ∀u ∈ P ∩ ∂Ω2

holds. Then T has a fixed point in .

To obtain the existence of positive solutions for system (2.1), we construct a cone
P in the Banach space Q = C(I, ℝ+) × C(I, ℝ+) equipped with the norm by

It is easy to see that P is a cone in Q.

Define two operators Tλ, Tμ : P → C(I, ℝ+) by

Then we can define an operator T : P → C(I, ℝ+) by

Lemma 2.6. Let (H1) hold. Then T : P → P is completely continuous.

Proof. Firstly, we prove that T : P → P. In fact, for all (u, v) ∈ P and t ∈ I, by Lemma 2.4(i) and (H1), it is obvious that Tλ(u, v)(t) > 0, Tμ(u, v)(t) > 0. In addition, we have

(2.18)

which implies . And we have

In a similar way,

Therefore,

This shows that T : P → P.

Secondly, we prove that T is continuous and compact, respectively. Let {(uk, vk)} ∈ P be any sequence of functions with ,

from the continuity of f, we know that ||Tλ(uk, vk) - Tλ(u, v)|| → 0 as k → ∞. Hence Tλ is continuous.

Tλ is compact provided that it maps bounded sets into relatively compact sets. Let , and let Ω be any bounded subset of P, then there exists r > 0 such that ||(u, v)|| ≤ r for all (u, v) ∈ Ω. Obviously, from (2.16), we know that

So, for all (u, v) ∈ Ω, TλΩ is equicontinuous. By Ascoli-Arzela theorem, we obtain that Tλ : P → P is completely continuous. In a similar way, we can prove that Tμ : P → P is completely continuous. Therefore, T : P → P is completely continuous. This completes the proof.

3 Main results

In this section, we will give our main results on multiplicity of positive solutions
of system (1.1). In the following, for convenience, we set

where qi(t), qj(t) ∈ Crd(I', ℝ+) satisfy

Theorem 3.1. Assume that (H1) holds. Assume further that

(H2) there exist a constant R > 0, and two functions pi(t) ∈ Crd(I, R+) satisfying such that

and one of the folloeing conditions is satisfied

(E1) , ,

(E2) , ,

(E3) , μ ∈ (0, N4),

(E4) λ ∈ (0, M4), ,

where

O1, O2 satisfy . Then system (1.1) has at least two positive solutions.

Proof. We only prove the case in which (H2) and (E1) hold, the other case can be proved similarly. Firstly, from (2.2), we have

From (3.2), (3.4), and (ii) of Lemma 2.5, it follows that system (2.1) has one positive solution (u1, v1) ∈ P with R2 ≤ ||(u1, v1)|| ≤ R1. Therefore, from Lemma 2.1, it follows that system (1.1) has one positive solution
(x1, y1). In the same way, from (3.2), (3.6), and (i) of Lemma 2.5, it follows that system (2.1) has one positive solution (u2, v2) ∈ P with R1 ≤ ||(u2, v2)|| ≤ R3. Therefore, from Lemma 2.1, it follows that system (1.1) has one positive solution
(x2, y2). Above all, system (1.1) has at least two positive solutions. This completes the
proof.

Then system (1.1) has at least two positive solutions for each and , where

Proof. We only prove the case in which (3.7) holds. The other case in which (3.8) holds
can be proved similarly.

Take

and let . For any t ∈ I, (u, v) ∈ ∂Ω4 ∩ P, it follows from λ > M5 and (H3) that

Consequently, for any (u, v) ∈ ∂Ω4 ∩ P, we have

(3.9)

From , , we know that , , we can choose ε3 > 0 such that M6 - ε3 > 0, N6 - ε3 > 0 and λf0 < M6 - ε3, μg0 < N6 - ε3. Then there exists such that for any and t ∈ I,

Take and . Then, for any (u, v) ∈ Ω5 ∩ P, from(3.1), we have

and

Consequently, for any (u, v) ∈ ∂Ω5 ∩ P, we have

(3.10)

From , we know that , , we can choose ε4 > 0 such that M6 - ε4 > 0, N6 - ε4 > 0 and λf∞< M6 - ε4, μg∞ < N6 - ε4. Then there exists such that for any and t ∈ I,

Take and let . Then, for any (u, v) ∈ Ω6 ∩ P, we have

and

Consequently, for any (u, v) ∈ ∂Ω6 ∩ P, we have

(3.11)

From (3.9), (3.10) and (i) of Lemma 2.5, it follows that system (2.1) has one positive solution (u1, v1) ∈ P with . Therefore, from Lemma 2.1, it follows that system (1.1) has one positive solution
(x1, y1). In the same way, from (3.9), (3.11) and (ii) of Lemma 2.5, it follows that system (2.1) has one positive solution (u2, v2) ∈ P with . Therefore, from Lemma 2.1, it follows that system (1.1) has one positive solution
(x2, y2). Above all, system (1.1) has at least two positive solutions. This completes the
proof.